Problem: Complete the square to rewrite this expression in the form $(x + a)^2 + b$. $a$ and $b$ can be positive or negative. $ x^{2}+6x+31$
If we square the sum of $x$ and another number, we get $ \begin{align} (x + a)^2 &= (x + a)(x + a) \\ &= x^2 + {2a}x + a^2\end{align}$ The number multiplying the $x$ term is $2$ times the number that was added inside the square. In the problem we're trying to solve, the number multiplying the $x$ term is $6$ , so when we rewrite the expression, the number added inside the square will be half of $6$ . (Which is $3$ $ (x + 3)^2 + b$ How can we find the value of $b$ If we multiply out the square in this expression, we get $ \begin{align} (x + 3)^2 + b &= (x + 3)(x + 3) + b \\ &= x^2 + 6x + 9 + b \end{align}$ This looks just like the given expression if $ 9 + b = 31$ So $b$ must be $22$ $x^{2}+6x+31$ can be rewritten as: $ (x + 3)^2 + 22$ Now it's just a square minus another number! We have completed the square.